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To see that these are reasonable, consider for example the
second, ab = a. This states that if a is included in &, then every-
thing that is hi both a and b is the whole of a.
From the stated postulates the following theorems on inehi-
sion (with thousands of more complicated ones, if desired) can
be proved. The specimens selected all agree with our intuitive
conception of what ^inclusion' means.
(1)  a < a.
(2)  lfa<b and b < c, then a < c.
(3)  Ifa<b and b < a, then a = b.
(4)  Z < a (where Z is the element in II a - it is proved to be
the only element satisfying II a).
(5)  a < U (where U is the element in II b - likewise unique).
(6)  a < a + b-, and ifa<y and b <ys then a + b < y.
(7) ab < a; andifx < aandx < b, thenx < ab.
(8)  Ifx < a and x < a', then x = Z; andifa < y and a' < y,
then y = U.
(9)  If a < br is false, then there is at least one element x,
distinct from Z> such that x < a and x < b.
It may be of interest to observe that4 <' in arithmetic and
analysis is the symbol for less than'. Note that if a, &, e,... are
real numbers, and 2 denotes zero, then (2) is satisfied for this
interpretation of '<', and similarly for (4), provided a is posi-
tive; but that (1) is not satisfied, nor is the second part of (6) -
as we see from 5 < 10,7 < 10, but 5 4- 7 < 10 is false.
The tremendous power and fluent ease of the method can be
readily appreciated by seeing what it does in any work on
symbolic logic. But, as already emphasized, the importance of
this 'symbolic reasoning* is hi its applicability to subtle ques-
tions regarding the foundations of all mathematics which, were
it not for this precise method of fixing meanings of 'words' or
other 'symbols* once for all, would probably be unapproachable
by ordinary mortals.
Like nearly all novelties, symbolic logic was neglected for
many years after its invention. As late as 1910 we find eminent
mathematicians scorning it as a 'philosophical' curiosity with-
out mathematical significance. The work of Whitehead and
Russell in Principia Mathematica (1910-13) was the first to